Question: Let $z$ be a complex number that satisfies
\[|z - 3i| + |z - 4| = 5.\]Find the minimum value of $|z|.$
Answer: By the Triangle Inequality,
\[|z - 3i| + |z - 4| = |z - 4| + |3i - z| \ge |(z - 4) + (3i - z)| = |-4 + 3i| = 5.\]But we are told that $|z - 3i| + |z - 4| = 5.$  The only way that equality can occur is if $z$ lies on the line segment connecting 4 and $3i$ in the complex plane.

[asy]
unitsize(1 cm);

pair Z = interp((0,3),(4,0),0.6);
pair P = ((0,0) + reflect((4,0),(0,3))*(0,0))/2;

draw((4,0)--(0,3),red);
draw((-1,0)--(5,0));
draw((0,-1)--(0,4));
draw((0,0)--Z);
draw((0,0)--P);
draw(rightanglemark((0,0),P,(4,0),8));

dot("$4$", (4,0), S);
dot("$3i$", (0,3), W);
dot("$z$", Z, NE);

label("$h$", P/2, NW);
[/asy]

We want to minimize $|z|$.  We see that $|z|$ is minimized when $z$ coincides with the projection of the origin onto the line segment.

The area of the triangle with vertices 0, 4, and $3i$ is
\[\frac{1}{2} \cdot 4 \cdot 3 = 6.\]This area is also
\[\frac{1}{2} \cdot 5 \cdot h = \frac{5h}{2},\]so $h = \boxed{\frac{12}{5}}.$